Rates of convergence in a central limit theorem for stochastic processes defined by differential equations with a small parameter

Abstract

Let μ be a positive finite Borel measure on the real line R. For t ≥ 0 let e t · E 1 and E 2 denote, respectively, the linear spans in L 2( R, μ) of { e isx , s > t} and { e isx , s < 0}. Let θ: R → C such that ∥θ∥ = 1, denote by α t ( θ, μ) the angle between θ · e t · E 1 and E 2. The problems considered here are that of describing those measures μ for which (1) α t ( θ, μ) > 0, (2) α t(θ, μ) → π 2 as t → ∞ (such μ arise as the spectral measures of strongly mixing stationary Gaussian processes), and (3) give necessary and sufficient conditions for the rate of convergence of the generalized maximal correlation coefficient: ϱ t ( θ, μ) = cos α t ( θ, μ). Using this coefficient we characterize the stationary continuous processes that are (a) completely regular and (b) strongly mixing Gaussian. We also give necessary and sufficient conditions for the rate of convergence of (a) the maximal correlation coefficient and (b) the mixing coefficient in the Gaussian case.